Iterating the Cesàro operators

Abstract

The discrete Cesàro operator C C associates to a given complex sequence s = { s n } s = \{s_n\} the sequence C s ≡ { b n } Cs \equiv \{b_n \} , where b n = s 0 + ⋯ + s n n + 1 , n = 0 , 1 , … b_n = \frac {s_0 + \dots + s_n}{n +1}, n = 0, 1, \ldots . When s s is a convergent sequence we show that { C n s } \{C^n s \} converges under the sup-norm if, and only if, s 0 = lim n → ∞ s n s_0 = \lim _{n\rightarrow \infty } s_n . For its adjoint operator C ∗ C^* , we establish that { ( C ∗ ) n s } \{(C^*)^n s\} converges for any s ∈ ℓ 1 s \in \ell ^1 . The continuous Cesàro operator, C f ( x ) ≡ 1 x ∫ 0 x f ( s ) d s C\!f (x) \ \equiv \ \frac {1}{x} \int _{0}^ {x}\, f(s) ds , has two versions: the finite range case is defined for f ∈ L ∞ ( 0 , 1 ) f \in L^\infty (0,1) and the infinite range case for f ∈ L ∞ ( 0 , ∞ ) f \in L^\infty (0, \infty ) . In the first situation, when f : [ 0 , 1 ] → C f: [0, 1] \rightarrow \mathbb {C} is continuous we prove that { C n f } \{C^n f \} converges under the sup-norm to the constant function f ( 0 ) f(0) . In the second situation, when f : [ 0 , ∞ ) → C f: [0, \infty )\rightarrow \mathbb {C} is a continuous function having a limit at infinity, we prove that { C n f } \{C^n f \} converges under the sup-norm if, and only if, f ( 0 ) = lim x → ∞ f ( x ) f(0) = \lim _{x\rightarrow \infty }f(x) .